In (1.3), the pointwise Caputo derivative \(d_t^{\alpha }\)  requires the
first-order derivative \(\frac{dv}{ds}\)  in any sense.
Therefore, in order to discuss \(d_t^{\alpha }v\)  for a function \(v\)  which keeps
apparently reasonable regularity such as "\(\alpha \) -times"
differentiability, we should formulate \(d_t^{\alpha }v\)  in a suitable
distribution space.
Moreover, such a formulation is not automatically unique.
There have been several works for example,
Kubica, Ryszewska and Yamamoto [1]}, Kubica and Yamamoto
[2]}, Zacher [3]}. Here we restrict ourselves to
an extremely limited number of references.
In this article, we extend the approach in Kubica, Ryszewska and
Yamamoto [1]} to discuss fractional derivatives in
fractional Sobolev spaces of arbitrary real number orders and
construct a convenient theory for
initial value problems and initial boundary value problems for
time-fractional differential equations.
In [1]}, the orders \(\alpha \)  is restricted to \(0<\alpha <1\) , but
here we study fractional orders \(\alpha \in \mathbb {R}\) .
